From Kant To Turing He Sun Max Planck Institute for Informatics

Question 1: Time vs. Space

Why randomness is useful in most settings, and ``should” be useless for poly-time algorithms? –Under reasonable assumptions BPP=P (1997). Why can hardness generate randomness? –One-Way Functions exist iff Pseudo-random Generators exist (1989). Randomness Evaluated by Computation Models = Randomness Evaluated by Entropy (1999) Question 2: Randomness in Computer Science x f(x) OWF f

Why Philosophers Should Care About Computational Complexity Scott Aaronson Computational Complexity and Turing Test Gödel’s idea on evolability and Leslie Valiant’s evolability theory PAC-Learning and the Problem of Induction Quantum Computing and the Many-Worlds Interpretation “Traditional ” Proofs and Zero-Knowledge Proofs Complexity, Space and Time

The Computer & The Brain, publications, 60 in pure mathematics, 30 in physics, and 60 in applied mathematics John von Neumann,

Scientific American, 1955 John George Kemeny,

Mind, 1950 Alan Turing,

Burke: Among the people listed above, who have great influence to your work and interests? Gödel: Only Kant is important. Grandjean Burke’s Questionnaire (1974)

Aristotle, 384 BC – 322 BC Metaphysics

Metaphysics began with the study of the knowledge of God and the nature of a future world. It was concluded early that good conduct would result in happiness in another world as arranged by God. Metaphysics

Immanuel Kant,

“The science of metaphysics must not attempt to reach beyond the limits of possible experience but must discuss only those limits, thus furthering the understanding of ourselves as thinking beings. The human mind is incapable of going beyond experience so as to obtain a knowledge of ultimate reality, because no direct advance can be made from pure ideas to objective existence. ” Critique of Pure Reason, 1781

What can I know? What should I know? What may I hope? What is a Human Being? Critique of Pure Reason, 1781

Friedrich Ludwig Gottlob Frege, Concept-Script: A Formal Language for Pure Thought Modeled on that of Arithmetic

1.There is a world that we need to describe. 2.We need some (mathematical, or other) language to express this world; 3.Question: Is what we say in this language the truth of the world? 4.We need to answer (3), by only using the chosen language. 5.Languages can never express the whole truth. Ludwig Josef Johann Wittgenstein, Tractatus Logico-philosophicus

Lowenheim-Skolem-Tarski Theorem, 1920

Gödel Incompleteness Theorem, 1931 Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In any sufficiently strong formal system there are true arithmetical statements that cannot be proved (in the system).

Church-Turing Thesis, 1935 Church: The notion of general recursive functions captures the informal idea of effective procedure. Turing: Whenever there is an effective method for obtaining the values of a mathematical function, the function can be computed by a Turing machine.

Since 1940s

Juris Hartmanis Notebook Entry on Dec : –“This was a good year.” This was a good 50 years. Philosophers and Computer Scientists have left more questions to each other. Thank You! Since 1960s